Racetovb4j

2022-02-03

The polynomial of degree 5, P(x) has leading coefficient 1, has roots of multiplicity 2 at x=1 and x=0, and a root of multiplicity 1 at x=-3, how do you find a possible formula for P(x)?

bubble53zjr

Expert

Since each root is a linear factor, we can write:
$P\left(x\right)={x}^{2}{\left(x-1\right)}^{2}\left(x+3\right)$
$={x}^{2}\left({x}^{2}-2x+1\right)\left(x+3\right)$
$={x}^{5}+{x}^{4}-5{x}^{3}+3{x}^{2}$
Any polynomial that contains these zeros and at least these multiplicities is going to be a multiple (scalar or polynomial) of this P(x)
$P\left(x\right)={x}^{5}+{x}^{4}-5{x}^{3}+3{x}^{2}$